Sunday, May 21, 2017

Is mathematics invention or discovery?

I think although natural numbers and few basic kinds of geometry, basic polynomials could be seen as inventions, on the other hand, real/complex/quaternion/octonion/sedenion numbers, decimal/hexadecimal/binary/octal number systems,
infinite family of arithmetic operations (addition, multiplication, power, tetration, ... and their inverses), fractal geometry, prime numbers etc all look like discoveries.
I think evidence for discovery is much more than evidence for invention but ultimately it maybe impossible to prove either side of the argument.
I think math is discovery and so math has its own existence but it is truly an abstract existence.
If math is an abstract existence then could any mathematical objects come into real existence by itself?
I think not.

Is mathematics infinite (when trivially infinite stuff taken out)?
For example it is trivial that each type of polynomial has infinite degrees (and dimensions (number of unknowns)) but is the total number of non-trivially different kinds of polynomials infinite?
What if we assume all kinds of possible polynomials as just one part of math?
Are the total number of such parts of math infinite?

Is Physics infinite?
In other words, how many non-trivially different universes mathematically possible that could support life/(human-like) intelligence?
I think it is obvious that if we change number of dimensions of the universe and found that universe could support life/intelligence
that universe must be counted as a non-trivially different universe
but what if we change one of basic constants of physics just a tiny bit, should we count that as a non-trivially different universe also? If not, then how much difference (as a percentage maybe)
for which basic constant of physics should be counted as a non-trivially different universe?
And all such different universes, which still following laws of physics of this universe, are the only possibilities?
What if we allow any kind of physical laws? How many non-trivially different sets of physical laws (for a universe that could support life/intelligence) possible?
(Of course, is it even theoretically/practically possible to mathematically determine if a given set of physical laws for a universe could support life/intelligence (when using computer simulations for experimentation included)?
How we could test if any given universe (set of physical laws) could support life and/or intelligence?
There is a an idea in computer science for testing equivalency.
For example it is known that all kinds of (completely different looking) NP-complete problems are actually equivalent because it is known how each one can be converted to one of the others.
Also it is known all kinds of theoretical computers are equivalent because all can be converted to a (Universal) Turing Machine.
Can we use the same idea for testing if any given universe is equivalent to our universe?
And if a universe is equivalent to ours, would not it mean that universe could also support life and intelligence?
Also there maybe other ways to test a universe for equivalency:
If we had a computer simulation of a (simplest) kind of life (living cells) then we could try to convert that simulation to use the physical laws of any given universe.
If we had a computer simulation of a (human-like) AI then we could try to convert that simulation to use the physical laws of any given universe, also.
And if we find that each simulation still works, would not it mean that universe could also support life and intelligence?
Also if what we trying to convert are computer simulations, what if we just design a (physical) computer in each universe we want to test? Wouldn't that be enough?)

Is computer science infinite?
(How many real/theoretical non-trivially different computer designs (hardware/software) possible?
Are all have equal power/ability (which is universal calculation)?)

Is chemistry infinite? (How many non-trivially different elements/molecules/chemical reactions possible?)

Is biology infinite? (How many non-trivially different species possible?)

I think, in a similar way, we could ask if any given science is infinite or not.
If any given science is infinite, is that mean humanity can never understand it as a whole?